Haikun Qi^{1}, Aurelien Bustin^{1}, Olivier Jaubert^{1}, RenĂ© Botnar^{1}, and Claudia Prieto^{1}

Free-running 3D radial (kooshball) sampling is suitable for fast and self-navigated whole-heart cardiovascular imaging. However, iterative undersampled 3D radial reconstruction requires computational demanding gridding/regridding steps in each iteration, which leads to long reconstruction time and may limit the applications of this imaging strategy. In this work, we investigate the feasibility of accelerating iterative reconstruction for a free-running 3D myocardial T1 mapping sequence using GRAPPA Operator Gridding (GROG)-based pre-reconstruction interpolation. Image quality and T1 estimation accuracy of the accelerated GROG-based reconstruction were compared with conventional non-uniform FFT (NUFFT)-based reconstruction in a standardized phantom and five healthy subjects.

*Pre-reconstruction Interpolation
using GROG:* The adopted free-running 3D myocardial T1
mapping sequence is shown in Fig. 1A. For retrospective reconstruction of T1
image series for a specific cardiac phase, the data
sorting process is shown in Fig. 1B. For a given cardiac phase, the highly undersampled 3D
radial data is reconstructed over multiple T1 contrasts by combining a dictionary-based
low-rank reconstruction with a recently proposed parallel imaging 3D
patch-based reconstruction, exploiting local, non-local and contrast
redundancies (8). The proposed
reconstruction is formulated in the following unconstrained Lagrangian:$$L\left(I,\alpha,Y\right)=argmin\parallel EI-\sqrt{D}K \parallel_2^2+\lambda\parallel \alpha \parallel_0+\mu\parallel I-P\alpha-Y \parallel_2^2$$where $$$E$$$ is the
encoding operator $$$E=\sqrt{D}AU_{r}FS$$$, including the non-Cartesian density compensation function $$$D$$$, sensitivity maps $$$S$$$, Fourier Transform $$$F$$$, low rank operator $$$U_{r}$$$, obtained by
truncating the singular value decomposition of the simulated dictionary, and the convolutional gridding operator $$$A$$$,
transforming Cartesian data back to 3D radial; $$$K$$$ is
the undersampled 3D radial data; $$$P$$$ is
the patch grouping operator and $$$\alpha$$$ are
the associated sparse coefficients; $$$Y$$$ is
the Lagrangian multiplier; $$$\lambda$$$ is the
sparsity regularization parameter and $$$\mu$$$ is
the penalty parameter. The above equation can be solved by operator-splitting
via alternating direction method of multipliers (ADMM) (8). For the
iterative minimization of the data consistency term $$$\parallel EI-\sqrt{D}K \parallel_2^2$$$, $$$E$$$ and
its Hermitian transpose $$$E^{H}$$$ involving
convolutional gridding needs to be performed in each iteration. Introducing GROG
by pre-interpolating $$$K$$$ into corresponding undersampled Cartesian
k-space $$$K_{c}$$$, the data consistency term is modified to
be $$$\parallel E_{GROG}I-WK_{c} \parallel_2^2$$$. The GROG-based
encoding operator is $$$E_{GROG}=WMU_{r}FS$$$,
where $$$M$$$ is the sampling mask of $$$K_{c}$$$, $$$W$$$ is the GROG weighting factor introduced to
resolve the blurring caused by the variable-density data distribution after
GROG (5) and is given by:$$W=\sqrt{D_{GROG,n}/D_{GROG,act}}$$where $$$D_{GROG,n}$$$ and $$$D_{GROG,act}$$$ are
the density filters given by the GROG algorithm using the actual number of
spokes and $$$n$$$ times the actual number of spokes, separately.
After reconstruction, T1 maps were generated by a dot product matching between
the reconstructed singular images and the dictionary.

*Data Acquisition and Analysis:* Imaging was
performed in a standardized T1 phantom and five heathy subjects on a 1.5T scanner
(Ingenia, Philips Healthcare). Imaging parameters of the free-running 3D radial
sequence were: FOV=200mm^{3}, spatial resolution=1.5mm^{3},
TR/TE/flip angle=11.6ms/5.1ms/6°, scan time=9.5min. Phantom T1 mapping accuracy
by non-uniform FFT (NUFFT)-based and GROG-based reconstruction was compared against
gold standard 2D inversion recovery spin echo (IR-SE) acquisition. For in vivo
analysis, T1 images were reconstructed at systolic cardiac phase. The
structural similarity index (SSIM) of the singular images reconstructed by the
two methods was measured. Three short-axis slices were selected from each
subject, and mean myocardium T1 was calculated for each slice and compared
between the two reconstruction methods by paired t-test. Five ADMM iterations
were performed for NUFFT-based and GROG-based reconstructions, with other
reconstruction parameters optimized separately for each method. All
reconstructions were performed on a server with dual 16-core CPUs, 256GB RAM.

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Fig. 1. A: The adopted free-running 3D whole-heart T1 mapping sequence,
which is IR (inversion recovery)-prepared, followed by continuous 3D golden
angle radial sampling with fast low angle shot. IR repetitions (IRTR) is
2200ms. B: Data sorting process for reconstruction of multiple T1 contrasts for
a given cardiac phase, which includes three steps: respiratory and cardiac
motion extraction; translational respiratory motion correction; binning into
different T1 contrasts and cardiac phases.

Fig.
2. Singular images and the corresponding T1 maps reconstructed from undersampled
3D radial data by direct NUFFT (A) and GROG interpolation with different
density filters, *D*_{GROG, act}
(B) and *D*_{GROG, 50} (C). Good
performance trade-off between SNR and blurring was achieved with* D*_{GROG, 50}.

Fig. 3. A: Phantom T1 mapping results by NUFFT-based and GROG-based
reconstructions. B: Mean and standard deviation of T1 values in each tube
estimated by the two reconstruction methods and comparison with gold standard 2D
IR-SE.

Fig. 4. Three short-axis T1 maps of a representative healthy subject reconstructed
by NUFFT-based (A) and GROG-based (B) methods.

Table 1: Comparison between NUFFT-based and GROG-based reconstruction.